= Solution
In a dilute solution, separate coils diffuse through solvent with long-ranged hydrodynamic coupling among their segments. The <Zimm model> gives whole-chain diffusion
$$
\boxed{D_{\rm dilute}\sim\frac{k_BT}{\eta_sR_H}\propto N^{-\nu},\qquad\nu\simeq3/5,}
$$
where $R_H$ is a <hydrodynamic radius> and $\eta_s$ the solvent <viscosity>. The coil continually changes shape while its <centre of mass> performs long-time <Brownian motion>; treating it as a rigid sphere is only a scaling analogy.
At high concentration, solvent-mediated <hydrodynamic interactions> are screened on sufficiently large scales and <times>. If chains are not entangled, additive local or blob friction gives <Rouse model> behaviour, $D\propto N^{-1}$ at fixed concentration. For long entangled chains, neighboring chains prevent rapid transverse escape. In the <reptation> tube picture the chain moves mainly along its contour until it renews the surrounding tube. Contour diffusion scales as $N^{-1}$, the tube-renewal <time> as $N^3$, and the random-coil size squared as $N$. Therefore
$$
\boxed{D_{\rm entangled}\sim\frac{R^2}{\tau_d}\propto N^{-2}.}
$$
Both cases ultimately give diffusive centre-of-mass motion, but entangled concentrated chains move much more slowly and retain constraints for longer. The inequality $\phi\gg\phi^*$ establishes strong overlap, not by itself entanglement: the latter also requires sufficiently many segments relative to an entanglement <length>. No numerical entanglement scale or concentrated-solution <diffusion coefficient> is supplied, so only the conditional regimes and scaling exponents can be fixed here.
Back to article page
